ELG350 Aigmt 3 Aigmt 3: E5.7; P5.6; P5.6; P5.9; AP5.; DP5.4 E5.7 A cotrol ytm for poitioig th had of a floppy dik driv ha th clodloop trafr fuctio 0.33( + 0.8) T ( ) ( + 0.6)( + 4 + 5) Plot th pol ad zro of th ytm ad dicu th domiac of th complx pol. What ovrhoot for a tp iput do you xpct? Solutio Th pol-zro map i how i Figur E5.7a. Sic th domiat pol ar ral, you do ot xpct to hav a larg ovrhoot, a how i Figur E5.7b. Figur 5.7a
Figur 5.7b P5.6 A robot i programmd to hav a tool or wldig torch follow a prcribd path. Coidr a robot tool that i to follow a awtooth path, a how i Fig. P5.6(a) i th txtbook. Th trafr fuctio of th plat i G ( ) 50( + ) ( + 6)( + 9) for th clod-loop ytm how i Fig. 5.6(b) i th txtbook. Calculat th tady-tat rror. Solutio W hav th ramp iput, r(t)t. So ad v 50( + ) 50 lim G( ) lim 0.96 0 0 ( 6)( 9) + + 54 R v 0.96.08
P5.6 Th trafr fuctio of th hart pump ad th pacmakr i foud to b G ( ) ( / + ) Dig th amplifir gai to yild a ytm with a ttlig tim to a tp diturbac of l tha c. Th ovrhoot to a tp i a dird hart rat hould b l tha 0%. (a) Fid a uitabl rag of. (b) If th omial valu of i 0, fid th itivity of th ytm to mall chag i. (c) Evaluat th itivity of part (b) at DC (t 0). (d) Evaluat th magitud of th itivity at th ormal hart rat of 60 bat/miut. Solutio (a) Th clod-loop trafr fuctio i T ) ( + + Th prct ovrhoot pcificatio P.O. 0% impli ζ 0. 59. From th charactritic quatio w fid that Solvig for yild ω ad ζω 6 (0.59) which impli that 8.6 So, ay gai i th itrval 0 < < 8.6 i valid. Th ttlig tim i T 4 / ζω 4 / 6 cod ad atifi th rquirmt. Notic that T i ot a fuctio of. (b) Th itivity i wh 0. (c) Th itivity at DC (0) ) + G( ) S T ( T S ( 0) 0 ( + ) + + 0
(d) I thi ca of 60 bat/miut, jπ. bat/c j π. So, th itivity at jπ i S T 85.084 ( jπ ) 0.77 0.3 P5.9 A ytm i how i Fig. P5.9 i th txtbook. (a) Dtrmi th tady-tat rror for a uit tp iput i trm of ad, whr E()R()-Y(). (b) Slct o that th tady-tat rror i zro. Solutio Th tady-tat rror i ( + 0)( + ) + ( ) 0 + ( ) lim 0 ( + 0)( + ) + 0 + To achiv a zro tady-tat trackig rror, lct a follow + 0 AP5. A clod-loop trafr fuctio i Y ( ) 96( + 3) T ( ) R( ) ( + 8)( + 8 + 36) (a) Dtrmi th tady-tat rror for a uit tp iput R()/. (b) Aum that th complx pol domiat, ad dtrmi th ovrhoot ad ttlig tim to withi % of th fial valu. (c) Plot th actual ytm rpo, ad compar it with th timat of part(b). Solutio (a) Th tady-tat rror i 96(3) lim ( T ( )) R( ) T (0) 0 0 8(36) (b) Aum th complx pol ar domiat. Th, w comput
a 0.75 ζω ic a3, ζ 0. 67 ad ω 6. Uig Figur 5.3 i Dorf & Bihop, w timat th ttlig tim ad prct ovrhoot to b 4 P.O. 45% ad T cod ζω (c) Th tp rpo i how i Figur AP5.. Th actual ttlig tim ad prct ovrhoot ar P.O. 3.4% ad T 0. 937 cod Figur AP5. DP5.4 Th pac atllit how i Fig. DP5.4(a) i th txtbook u a cotrol ytm to radjut it oritatio, a how i Fig. DP5.4(b) i th txtbook. (a) Dtrmi a cod-ordr modl for th clod-loop ytm. (b) Uig th cod-ordr modl, lct a gai o that th prct ovrhoot i l tha 5% ad th tady-tat rror to a tp i l tha %. (c) Vrify your dig by dtrmiig th actual prformac of th third-ordr ytm.
Solutio Th op-loop trafr fuctio i 0 GG c ( ) ( + 90)( + )( + 9) 0 / 90 ( / 90 + )( + )( + 9) Th cod-ordr approximatio i obtaid by glctig th fatt firt-ordr pol. Thu, / 9 GG c ( ) ( + )( + 9) Th clod-loop trafr fuctio i T ( ) / 9 + 0 + 9 + / 9 Wh ζ 0. 5, w hav l tha 5% ovrhoot. So, w hav ad ξω 0 ω + 9 / 9 Elimiatig ω ad olvig for (with P.O. 5%) yild Alo, ad 75 limgg 0 0 ( ) 90(9) p c 8 + p < 0. + / 8 impli 594
Thrfor, 594 75. Uig th cod-ordr modl, w ca lct 75 to mt both pcificatio. Th actual ovrhoot i P.O. 6% ad th tady-tat rror i 9.7%. Lowrig th gai to 63 will atify both pcificatio.